LMIs in Control/pages/Robust Stabilization of Second-Order Systems

LMIs in Control/pages/Robust Stabilization of Second-Order Systems

Stabilization is a vastly important concept in controls, and is no less important for second order systems with perturbations. Such a second order system can be conceptualized most simply by the model of a mass-spring-damper, with added perturbations. Velocity and position are of course chosen as the states for this system, and the state space model can be written as it is below. The goal of stabilization in this context is to design a control law that is made up of two controller gain matrices $K_p$, and $K_d$. These allow the construction of a stabilized closed loop controller.

The System

In this LMI, we have an uncertain second-order linear system, that can be modeled in state space as:

$\begin{array}{cc}(A_2+\mathrm{\Delta }{A}_2)\ddot{x}+(A_1+\mathrm{\Delta }{A}_1)\dot{x}+(A_0+\mathrm{\Delta }{A}_0)x& =Bu)\\ y_d& =C_d\dot{x}\\ y_p& =C_{p}x\end{array}$

where $x\in R^n$ and $u\in R^r$ are the state vector and the control vector, respectively, $y_d\in R^{m_p}$ and $y_d\in R^{m_p}$ are the derivative output vector and the proportional output vector, respectively, and $A_2,A_1,A_0,B,C_d,C_p$ are the system coefficient matrices of appropriate dimensions.

$\mathrm{\Delta }{A}_2,\mathrm{\Delta }{A}_1,$ and $\mathrm{\Delta }{A}_0$ are the perturbations of matrices $A_2,A_1,$ and $A_0$, respectively, are bounded, and satisfy

$\begin{array}{c}|\mathrm{\Delta }{A}_2{|}_2\le {ϵ}_2,|\mathrm{\Delta }{A}_1{|}_2\le {ϵ}_1,|\mathrm{\Delta }{A}_0{|}_2\le {ϵ}_0,\end{array}$

or

$\begin{array}{c}max\{\Vert \mathrm{\Delta }{a}_{2ij}\Vert \}\le {\eta }_2,max\{\Vert \mathrm{\Delta }{a}_{1ij}\Vert \}\le {\eta }_1,max\{\Vert \mathrm{\Delta }{a}_{0ij}\Vert \}\le {\eta }_0,\end{array}$

where ${ϵ}_2,{ϵ}_1,{ϵ}_0$ and ${\eta }_2,{\eta }_1,{\eta }_0$ are two sets of given positive scalars, $\mathrm{\Delta }{a}_{2ij},\mathrm{\Delta }{a}_{1ij},$ and $\mathrm{\Delta }{a}_{0ij}$ are the i-th row and j-th collumn elements of matrices $\mathrm{\Delta }{A}_2,\mathrm{\Delta }{A}_1,$ and $\mathrm{\Delta }{A}_0,$, respectively. Also, the perturbation notations also satisfy the assumption that $\mathrm{\Delta }{A}_2,\mathrm{\Delta }{A}_0\in S^n$ and $A_2+\mathrm{\Delta }{A}_2>0$.

To further define: $x$ is$\in R^n$ and is the state vector, $A_0$ is $\in R^{n*n}$ and is the state matrix on $x$ , $A_1$ is $\in R^{n*n}$ and is the state matrix on $\dot{x}$ , $A_2$ is $\in R^{n*n}$ and is the state matrix on $\ddot{x}$, $B$ is $\in R^{n*r}$ and is the input matrix, $u$ is $\in R^r$ and is the input, $C_d$ and $C_p$ are $\in R^{m*n}$ and are the output matrices, $y_d$ is $\in R^m$ and is the output from $C_d$, and $y_p$ is $\in R^m$ and is the output from $C_p$.

The Data

The matrices $A_2,A_1,A_0,B,C_d,C_p$ and perturbations $\mathrm{\Delta }{A}_2,\mathrm{\Delta }{A}_1,\mathrm{\Delta }{A}_0,$ describing the second order system with perturbations.

The Optimization Problem

For the system defined as shown above, we need to design a feedback control law such that the following system is Hurwitz stable. In other words, we need to find the matrices $K_p$ and $K_d$ in the below system.

$\begin{array}{cc}(A_2+\mathrm{\Delta }{A}_2)\ddot{x}+(A_1-B{K}_p{C}_p+\mathrm{\Delta }{A}_1)\dot{x}+(A_0-B{K}_d{C}_d+\mathrm{\Delta }{A}_0)x& =0\\ \end{array}$

However, to do proceed with the solution, first we need to present a Lemma. This Lemma comes from Appendix A.6 in "LMI's in Control systems" by Guang-Ren Duan and Hai-Hua Yu. This Lemma states the following: if $A_2>0,A_1+A_1^T>0,A_0>0$, then the following is also true for the system described above:

The system is hurwitz stable if

$\begin{array}{c}{\lambda }_{min}(A_2)>|\mathrm{\Delta }{A}_2{|}_2,{\lambda }_{min}(A_1+A_1^T)>|\mathrm{\Delta }{A}_1{|}_2,{\lambda }_{min}(A_0)>|\mathrm{\Delta }{A}_0{|}_2\end{array}$,

or

the system is hurwitz stable if

$\begin{array}{c}{\lambda }_{min}(A_2)>\sqrt{l_2}max\{\Vert \mathrm{\Delta }{a}_{2ij}\Vert \},{\lambda }_{min}(A_1+A_1^T)>\sqrt{l_1}max\{\Vert \mathrm{\Delta }{a}_{1ij}\Vert \},{\lambda }_{min}(A_0)>\sqrt{l_0}max\{\Vert \mathrm{\Delta }{a}_{0ij}\Vert \}\end{array}$

, where $l_2,l_1,l_0$ are the numbers of nonzero elements in matrices $\mathrm{\Delta }{A}_2,\mathrm{\Delta }{A}_1,\mathrm{\Delta }{A}_0,$ respectively.

The LMI: Robust Stabilization of Second Order Systems

This problem is solved by finding matrices $K_p\in R^{r*m_p}$ and $K_d\in R^{r*m_d}$ that satisfy either of the following sets of LMIs.

$\begin{array}{cc}{A}_0-B{K}_d{C}_d& >{ϵ}_{0}I,\\ (A_1-B{K}_p{C}_p)+(A_1-B{K}_p{C}_p{)}^T& >{ϵ}_{1}I.\end{array}$

or

$\begin{array}{cc}{A}_0-B{K}_d{C}_d& >{\eta }_0\sqrt{l_0}I,\\ (A_1-B{K}_p{C}_p)+(A_1-B{K}_p{C}_p{)}^T& >{\eta }_1\sqrt{l_1}I.\end{array}$

Conclusion:

Having solved the above problem, the matrices $K_p$ and $K_d$ can be substituted into the input as $u=K_p{C}_p\dot{x}+K_d{C}_{d}x$ to robustly stabilize the second order uncertain system. The new system is stable in closed loop.

Implementation

This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/ROBstab2ndorder.m

Related LMIs

Stabilization of Second-Order Systems

External Links

This LMI comes from

• [1] - "LMIs in Control Systems: Analysis, Design and Applications" by Guang-Ren Duan and Hai-Hua Yu

Other resources:

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

References

Duan, G. (2013). LMIs in control systems: analysis, design and applications. Boca Raton: CRC Press, Taylor & Francis Group.